Determine the roughness-controlled-layer thickness for a densely packed oscillatory bed

Determine the thickness of the roughness-controlled layer produced by a single layer of spheres resting on a wall under oscillatory flow, and establish how the minimum element spacing affects that thickness relative to steady-flow estimates.

Background

Steady-flow studies commonly place the roughness-controlled layer two to five element heights above roughness crests. The paper notes that this estimate is based on canopy and sparse-array geometries, whereas a single layer of spheres resting on a wall has no permeable canopy interior and has the smallest geometrically permitted element spacing.

Because oscillatory forcing reverses every half-period, wake development may not follow the steady-flow picture. Resolving the thickness of the oscillatory roughness-controlled layer is necessary to explain how a logarithmic profile can coexist with densely packed gravel-scale roughness.

References

What the steady-flow picture leaves open is the thickness. The estimate eq:jimenez rests on canopy and sparse-array geometries; a single layer of spheres resting on a wall has no canopy interior, and its spacing is the smallest the geometry permits. What that packing implies for the layer thickness is the subject of \S~\ref{sec:kinematic decay}.

— Roughness-controlled layer in oscillatory turbulent boundary layers over densely packed uniform roughness  (2608.17543 - Liu et al., 18 Aug 2026) in Theoretical background, Section 2.1, “The roughness-controlled layer in steady flow”